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Alan Fern
* Based in part on slides by Daniel Weld
Reinforcement Learning Active Learning
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Active Reinforcement Learning
So far, we’ve assumed agent has a
policy
We just learned how good it is
Now, suppose agent must learn a good
policy (ideally optimal)
While acting in uncertain world
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Naïve Model-Based Approach
1. Act Randomly for a (long) time
Or systematically explore all possible actions
2. Learn
Transition function
Reward function
3. Apply value/policy iteration to get policy
4. Follow resulting policy thereafter.
Will this work?
Any problems?
Yes (if we do step 1 long enough and
there are no “dead-ends”)
We will act randomly for a long time
before exploiting what we know.
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Revision of Naïve Approach
1. Start with initial (uninformed) model
2. Solve for optimal policy given current model
(using value or policy iteration)
3. Execute action suggested by policy in current state
4. Update estimated model based on observed transition
5. Goto 2
This is just ADP but we follow the greedy policy suggested by
current value estimate
Will this work?
No. Can get stuck in local minima.
(depends on initialization)
What can be done?
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Two reasons to take an action in RL Exploitation: To try to get reward. We exploit our
current knowledge to get a payoff.
Exploration: Get more information about the world. How do we know if there is not a pot of gold around the corner.
To explore we typically need to take actions that do not seem best according to our current model.
Managing the trade-off between exploration and exploitation is a critical issue in RL
Basic intuition behind most approaches: Explore more when knowledge is weak
Exploit more as we gain knowledge
Exploration versus Exploitation
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ADP-based (model-based) RL
1. Start with initial model
2. Solve for optimal policy given current model
(using value or policy iteration)
3. Take action according to an explore/exploit policy
(explores more early on and gradually uses policy from 2)
4. Update estimated model based on observed transition
5. Goto 2
This is just ADP but we follow the explore/exploit policy
Will this work?
Depends on the explore/exploit policy.
Any ideas?
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Explore/Exploit Policies Greedy action is action maximizing estimated Q-value
where V is current optimal value function estimate (based on current model), and R, T are current estimates of model
Q(s,a) is the expected value of taking action a in state s and then getting the estimated value V(s’) of the next state s’
Want an exploration policy that is greedy in the limit of infinite exploration (GLIE) Guarantees convergence
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Explore/Exploit Policies GLIE Policy 1
On time step t select random action with probability p(t) and greedy action with probability 1-p(t)
p(t) = 1/t will lead to convergence, but can be slow
In practice it is common to simply set p(t) to a small constant ε (e.g. ε=0.1) Called ε-greedy exploration
Just as we saw for bandits
ε usually set to small value (compared to 0.5) so the trajectories we learn from are mostly based on exploitation behavior
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Explore/Exploit Policies GLIE Policy 2: Boltzmann Exploration
Select action a with probability,
T is the temperature. Large T means that each action has about the same probability. Small T leads to more greedy behavior.
Typically start with large T and decrease with time
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The Impact of Temperature
Suppose we have two actions and that Q(s,a1) = 1, Q(s,a2) = 2
T=10 gives Pr(a1 | s) = 0.48, Pr(a2 | s) = 0.52
Almost equal probability, so will explore
T= 1 gives Pr(a1 | s) = 0.27, Pr(a2 | s) = 0.73
Probabilities more skewed, so explore a1 less
T = 0.25 gives Pr(a1 | s) = 0.02, Pr(a2 | s) = 0.98
Almost always exploit a2
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Alternative Model-Based Approach:
Optimistic Exploration
There is a class of RL algorithms based on the idea of
optimistic exploration.
Basically, if the agent has not explored a state “enough”,
then it acts as if that state provides maximum reward
So actions will be selected to try and reach such states
Many of the theoretical results are based on this idea
We’ll only touch on the theory
We’ll cover two views of essentially the same approach
The first view from your book
The second view that is more standard in the literature
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Alternative Model-Based Approach:
Optimistic Exploration (view 1)
1. Start with initial model
2. Solve for “optimistic policy”
(uses optimistic variant of value iteration)
(inflates value of actions leading to unexplored regions)
3. Take greedy action according to optimistic policy
4. Update estimated model
5. Goto 2
Basically act as if all “unexplored” state-action pairs
are maximally rewarding.
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Optimistic Value Iteration Recall that value iteration iteratively performs the following update
at all states:
Optimistic variant adjusts update to make actions that lead to unexplored regions look good
Optimistic VI: assigns highest possible value Vmax to any state-action pair that has not been explored enough Maximum value is when we get maximum reward forever
What do we mean by “explored enough”? N(s,a) > Ne, where N(s,a) is number of times action a has been
tried in state s and Ne is a user selected parameter
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Optimistic Value Iteration
Optimistic value iteration computes an optimistic value function V+ using following updates
The agent will behave initially as if there were wonderful rewards scattered all over around– optimistic .
But after actions are tried enough times we will perform standard “non-optimistic” value iteration
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Standard VI
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Optimistic Exploration (view 1)
1. Start with initial model
2. Solve for optimistic policy using optimistic value iteration
3. Take greedy action according to optimistic policy
4. Update estimated model; Goto 2
Can any guarantees be made for the algorithm?
• If Ne is large enough and all state-action pairs are explored that many times, then the model will be accurate and lead to close to optimal policy
• But, perhaps some state-action pairs will never be explored enough or it will take a very long time to do so
• Optimistic exploration is equivalent to another algorithm, Rmax, which has been proven to efficiently converge
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Alternative View of Optimistic
Exploration: The Rmax Algorithm
1. Start with an optimistic model
(assign largest possible reward to “unexplored states”)
(actions from “unexplored states” only self transition)
2. Solve for optimal policy in optimistic model (standard VI)
3. Take greedy action according to policy
4. Update optimistic estimated model
(if a state becomes “known” then use its true statistics)
5. Goto 2
Agent always acts greedily according to a model that
assumes all “unexplored” states are maximally
rewarding
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Rmax: Optimistic Model
Keep track of number of times a state-action pair is tried A state is “unknown” if some action is not explored enough from it
If N(s,a) < Ne then T(s,a,s)=1 and R(s) = Rmax in optimistic model,
Otherwise T(s,a,s’) and R(s) are based on estimates obtained from the Ne experiences (the estimate of true model) For large enough Ne these will be accurate estimates
An optimal policy for this optimistic model will try to reach unexplored states (those with unexplored actions) since it can stay at those states and accumulate maximum reward
Never explicitly explores. Is always greedy, but with respect to an optimistic outlook.
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Optimistic Exploration Rmax is equivalent to optimistic exploration via optimistic
VI Convince yourself of this.
Is Rmax provably efficient? If the model is ever completely learned (i.e. N(s,a) > Ne, for
all (s,a), then the policy will be near optimal)
Recent results show that this will happen “quickly”
PAC Guarantee (Roughly speaking): There is a value of Ne (depending on n,m, and Rmax), such that with high probability the Rmax algorithm will select at most a polynomial number of action with value less than ε of optimal.
RL can be solved in poly-time in n, m, and Rmax!
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TD-based Active RL
1. Start with initial value function
2. Take action from explore/exploit policy giving new state s’
(should converge to greedy policy, i.e. GLIE)
3. Update estimated model
4. Perform TD update
V(s) is new estimate of optimal value function at state s.
5. Goto 2
Just like TD for passive RL, but we follow explore/exploit policy
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Given the usual assumptions about learning rate and GLIE,
TD will converge to an optimal value function!
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Requires an estimated model. Why?
TD-based Active RL
1. Start with initial value function
2. Take action from explore/exploit policy giving new state s’
(should converge to greedy policy, i.e. GLIE)
3. Update estimated model
4. Perform TD update
V(s) is new estimate of optimal value function at state s.
5. Goto 2
To compute the explore/exploit policy.
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TD-Based Active Learning Explore/Exploit policy requires computing Q(s,a) for the
exploit part of the policy Computing Q(s,a) requires T and R in addition to V
Thus TD-learning must still maintain an estimated model for action selection
It is computationally more efficient at each step compared to Rmax (i.e. optimistic exploration) TD-update vs. Value Iteration
But model requires much more memory than value function
Can we get a model-fee variant?
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Q-Learning: Model-Free RL
Instead of learning the optimal value function V, directly
learn the optimal Q function.
Recall Q(s,a) is the expected value of taking action a in state s and
then following the optimal policy thereafter
Given the Q function we can act optimally by selecting
action greedily according to Q(s,a) without a model
The optimal Q-function satisfies
which gives:
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How can we learn the Q-function directly?
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Q-Learning: Model-Free RL
We can perform updates after each action just like
in TD.
After taking action a in state s and reaching state s’ do:
(note that we directly observe reward R(s))
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(noisy) sample of Q-value
based on next state
Bellman constraints on optimal Q-function:
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Q-Learning 1. Start with initial Q-function (e.g. all zeros)
2. Take action from explore/exploit policy giving new state s’ (should converge to greedy policy, i.e. GLIE)
3. Perform TD update Q(s,a) is current estimate of optimal Q-function.
4. Goto 2
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Does not require model since we learn Q directly!
Uses explicit |S|x|A| table to represent Q
Explore/exploit policy directly uses Q-values
E.g. use Boltzmann exploration.
Book uses exploration function for exploration (Figure 21.8)
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Q-Learning: Speedup for Goal-Based Problems
Goal-Based Problem: receive big reward in
goal state and then transition to terminal state
Parking domain is goal based
Consider initializing Q(s,a) to zeros and then
observing the following sequence of (state,
reward, action) triples
(s0,0,a0) (s1,0,a1) (s2,10,a2) (terminal,0)
The sequence of Q-value updates would
result in: Q(s0,a0) = 0, Q(s1,a1) =0, Q(s2,a2)=10
So nothing was learned at s0 and s1
Next time this trajectory is observed we will get
non-zero for Q(s1,a1) but still Q(s0,a0)=0
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Q-Learning: Speedup for Goal-Based Problems
From the example we see that it can take
many learning trials for the final reward to
“back propagate” to early state-action pairs
Two approaches for addressing this problem:
1. Trajectory replay: store each trajectory and do
several iterations of Q-updates on each one
2. Reverse updates: store trajectory and do
Q-updates in reverse order
In our example (with learning rate and
discount factor equal to 1 for ease of
illustration) reverse updates would give
Q(s2,a2) = 10, Q(s1,a1) = 10, Q(s0,a0)=10
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Q-Learning: Suggestions for Q-
Learning Assignment
A very simple exploration strategy is -greedy
exploration (generally called “epsilon greedy”)
Select a “small” value for e (perhaps 0.1)
On each step:
With probability select a random action, and
with probability 1- select a greedy action
But it might be interesting to play with
exploration a bit (e.g. compare to a
decreasing exploration rate)
You can use a discount factor of one or close
to 1.
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Active Reinforcement Learning Summary
Methods
ADP
Temporal Difference Learning
Q-learning
All converge to optimal policy assuming a GLIE
exploration strategy
Optimistic exploration with ADP can be shown to
converge in polynomial time with high probability
All methods assume the world is not too
dangerous (no cliffs to fall off during exploration)
So far we have assumed small state spaces
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ADP vs. TD vs. Q
Different opinions.
(my opinion) When state space is small then this is not such
an important issue.
Computation Time
ADP-based methods use more computation time per step
Memory Usage ADP-based methods uses O(mn2) memory
Active TD-learning uses O(mn2) memory (must store model)
Q-learning uses O(mn) memory for Q-table
Learning efficiency (performance per unit experience)
ADP-based methods make more efficient use of experience by
storing a model that summarizes the history and then reasoning
about the model (e.g. via value iteration or policy iteration)
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What about large state spaces?
One approach is to map original state space S to a much smaller state space S’ via some lossy hashing function. Ideally “similar” states in S are mapped to the same state in S’
Then do learning over S’ instead of S. Note that the world may not look Markovian when viewed through
the lens of S’, so convergence results may not apply
But, still the approach can work if a good enough S’ is engineered (requires careful design), e.g.
Empirical Evaluation of a Reinforcement Learning Spoken Dialogue System. With S. Singh, D. Litman, M. Walker. Proceedings of the 17th National Conference on Artificial Intelligence, 2000
We will now study three other approaches for dealing with large state-spaces Value function approximation
Policy gradient methods
Least Squares Policy Iteration
